Calculate the implied Par Yield from the Theoretical Zero-Coupon Spot Rate Curve.
2 Years
3 Years
4 Years
5 Years
Please enter valid numeric values for all required spot rates.
Calculated Par Yield
0.00%
function toggleInputs() {
var term = parseInt(document.getElementById('maturityTerm').value);
// Reset visibility
document.getElementById('spot3').parentElement.style.visibility = 'visible';
document.getElementById('spot4').parentElement.style.visibility = 'visible';
document.getElementById('spot5').parentElement.style.visibility = 'visible';
// Hide based on selection
if (term < 5) document.getElementById('spot5').parentElement.style.visibility = 'hidden';
if (term < 4) document.getElementById('spot4').parentElement.style.visibility = 'hidden';
if (term < 3) document.getElementById('spot3').parentElement.style.visibility = 'hidden';
}
function calculateParRate() {
// Hide previous results/errors
document.getElementById('resultBox').style.display = 'none';
document.getElementById('errorMsg').style.display = 'none';
var term = parseInt(document.getElementById('maturityTerm').value);
var spots = [];
var isValid = true;
// Gather Inputs
for (var i = 1; i <= term; i++) {
var val = parseFloat(document.getElementById('spot' + i).value);
if (isNaN(val)) {
isValid = false;
break;
}
spots.push(val / 100); // Convert percentage to decimal
}
if (!isValid) {
document.getElementById('errorMsg').style.display = 'block';
return;
}
// Calculation Logic
// Formula: Par Rate (n) = (1 – DF_n) / Sum(DF_1 to DF_n)
// DF_t = 1 / (1 + z_t)^t
var sumDF = 0;
var finalDF = 0;
var dfArray = [];
for (var t = 0; t < spots.length; t++) {
var year = t + 1;
var spotRate = spots[t];
// Calculate Discount Factor for year t
var df = 1 / Math.pow((1 + spotRate), year);
dfArray.push(df);
sumDF += df;
// Store the final DF for the numerator
if (year === term) {
finalDF = df;
}
}
// Calculate Par Rate
var numerator = 1 – finalDF;
var denominator = sumDF;
var parRateDecimal = numerator / denominator;
var parRatePercent = (parRateDecimal * 100).toFixed(4);
// Display Result
document.getElementById('resultBox').style.display = 'block';
document.getElementById('parResult').innerHTML = parRatePercent + '%';
// Display Breakdown
var breakdownHtml = 'Calculation Breakdown:';
breakdownHtml += 'Numerator (1 – DF' + term + '): ' + numerator.toFixed(5) + ";
breakdownHtml += 'Denominator (∑ DF): ' + denominator.toFixed(5) + ";
breakdownHtml += 'Implied ' + term + '-Year Par Bond Coupon: ' + parRatePercent + '%';
document.getElementById('breakdown').innerHTML = breakdownHtml;
}
// Initialize visibility on load
toggleInputs();
How to Calculate Par Rate from Spot Rate
In fixed-income analytics, understanding the relationship between the Spot Rate (zero-coupon yield) and the Par Rate (yield to maturity of a coupon-paying bond priced at par) is crucial for bond pricing and valuation. This calculator allows you to derive the Par Rate curve from a given Spot Rate curve using the principle of "bootstrapping" or discount factor summation.
What is the Difference?
Spot Rate ($z_t$): The yield on a zero-coupon bond for a specific maturity $t$. It represents the interest rate for a single cash flow at a specific future date.
Par Rate ($C_n$): The coupon rate that causes a bond's price to equal its face value (par) today. It is essentially a weighted average of the spot rates covering the bond's life.
The Mathematical Formula
To calculate the Par Rate for maturity $n$, we equate the price of the bond (100% of par) to the present value of its cash flows (coupons and principal), discounted using the respective spot rates for each period.
The formula for the Par Rate ($C_n$) is derived as follows:
$$ C_n = \frac{1 – DF_n}{\sum_{t=1}^{n} DF_t} $$
Where:
$n$ = Maturity year.
$DF_t$ = Discount Factor for year $t$, calculated as $1 / (1 + z_t)^t$.
$z_t$ = Spot rate for year $t$.
Step-by-Step Calculation Example
Let's calculate the 3-Year Par Rate given the following spot rate curve:
Year 1 Spot Rate: 3.0%
Year 2 Spot Rate: 3.5%
Year 3 Spot Rate: 4.0%
Step 1: Calculate Discount Factors (DF)
Year ($t$)
Spot Rate ($z_t$)
Formula ($1 / (1+z_t)^t$)
Discount Factor ($DF_t$)
1
3.0% (0.03)
$1 / (1.03)^1$
0.97087
2
3.5% (0.035)
$1 / (1.035)^2$
0.93351
3
4.0% (0.04)
$1 / (1.04)^3$
0.88900
Step 2: Sum the Discount Factors
Sum = $0.97087 + 0.93351 + 0.88900 = \mathbf{2.79338}$
Result: The 3-Year Par Rate is approximately 3.97%. Notice how this is slightly lower than the 3-year spot rate (4.0%) because the earlier cash flows (coupons in years 1 and 2) are discounted at lower rates (3.0% and 3.5%).
Why is this Important?
1. Bond Valuation: Most bonds pay coupons. To price a new coupon-paying bond correctly, issuers need to know the par rate to set the coupon so the bond sells at par value.
2. Arbitrage Opportunities: If the actual market par yield differs significantly from the par yield implied by the spot curve, traders may find arbitrage opportunities by buying/selling the bond and stripping/reconstituting it into zero-coupon instruments.
3. Curve Construction: This math is reversible. Often, the market gives us Par Rates (from actively traded government bonds), and we must use "Bootstrapping" to derive the theoretical Spot Rate curve.